Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering 43: Electrical Circuits & Devices
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

BackGround EE ( §5.5)
 Consider Sinusoidal Sources
V s of the form

V m cos
  t
  v
 or I s

I m cos
  t
  i

 Alternatively
V s

V rms cos
  t
  v
 or I s

I rms cos
  t
  i

 Recall for a Sinusoid
U rms

U m
2 or
 Complex Power
U m

2

U rms
S

P
 jQ
• Where
– S ≡ Complex Power in V∙A
– P ≡ Real Power in Watts
– Q ≡ ReActive Power in VAR
Engineering 43: Electrical Circuits & Devices
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

BackGround EE ( §5.5)
 Each Passive Component
(R, C, L, V src
, I src
) Generates it’s own Complex Power in an operating Circuit
 Furthermore Complex Power is Conserved, just a real power was conserved in purely resistive Circuits; i.e.: k n 

1
S k

0 for the " n" Components
 Equating Real & Imag Parts

P k

0 &

Q k

0
Engineering 43: Electrical Circuits & Devices
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

R, C, L vs. P&Q
4
 Note that the passive sign
Convention Applies to
Complex Power a well
• Positive Pwr → ABSORBED or
DISSIPATED
• Negative Pwr → SUPPLIED or
P
R
I
2 rms
R
GENERATED
 RESISTORS can only
DISSIPATE Real Power per
Text Eqns (5.69) & (5.71)
P
R

I
2
Rrms
R & P
R
 2
V
Rrms
R
• I
Rrms
≡ Current thru R
• V
Rrms
≡ Voltage across R
Engineering 43: Electrical Circuits & Devices Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

R, C, L vs. P&Q
 INDUCTORS create
POSITIVE ReActive
(imaginary) Power per Text
Eqns (5.70) & (5.72)
Q
L
 2
I
Lrms
X
L
& Q
L
 2
V
Lrms
X
L
• Where
– I
Lrms
– X
L
≡ Current thru L
– V
Lrms
≡ Voltage across L
≡ Inductive ReActance, ωL
Engineering 43: Electrical Circuits & Devices
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

R, C, L vs. P&Q
 CAPACITORS create
NEGATIVE ReActive
Q
C
(imaginary) Power per Text
Eqns (5.70) & (5.72)

I
2
Crms
 
C
& Q
C
 2
V
Crms
 
C or
Q
C
 2
I
Crms
1

C
• Where
& Q
C
 2
V
Crms


– I
Crms
≡ Current thru C
– V
Crms
≡ Voltage across C
– X
C
≡ Capacitive ReActance,
−1/(ωC)
1
  
Engineering 43: Electrical Circuits & Devices
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

S by ShortHand Calc
 Alternatively Calculate S for any Component using the
COMPLEX CONJUGATE of the CURRENT
Thru the device
S

1
2
VI
*
 Recall Conjugation for
Imaginary of Phasor Values
U

U m
  
U
* 
U m

  or
U

A
 jB

U
* 
A
 jB
Engineering 43: Electrical Circuits & Devices
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

S by ShortHand Calc
 If S

1
2
P k
VI
*

Re
  k

 Then
Q k

Im
  k

 Also For Sources the
Re



1
2
V k
I
* k 


Im



1
2
VI
* k 


Apparent Power :
P apparent

V
Srms
I
Srms

S

1
2
V
S
I
S
* as
1
2
V
S
I
S
* 
V
S
2
I
S
*
2

V
S
2

I
S
*
2 and
V
S
2

I
S
*
2

V
S
2

I
S
*
2

V
Sm
2
Engineering 43: Electrical Circuits & Devices
I
Sm
2

V
Srms
I
Srms
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx
8

Apparent Power
 Apparent Power Calculations are usually used with
SOURCES, but the concept also applies to Impedance
Loads
 Given an Impedance with
Current I
Z
, and Voltage-Drop
V
Z then
P apparent

V
Zrms
I
Zrms

S
Z

1
2
V
Z
I
Z
* also since S
Z

P
Z
 jQ
Z
P apparent

P
Z
2 
Q
Z
2 
S
Z
Engineering 43: Electrical Circuits & Devices
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Problem 5.80
 Given Circuit
10
 Find The Power Condition for each element, the Quantity,
Nature, and if the element is
Absorbing/Supplying Power
 Note that this is SINGLE
LOOP Circuit which thus has
ONE Current
Engineering 43: Electrical Circuits & Devices Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Confirm: ∑P avg
= 0
 Note that the Average
Powers Add to ZERO as they must by conservation of
Energy
Engineering 43: Electrical Circuits & Devices
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Check ReActive Power
Engineering 43: Electrical Circuits & Devices
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx

Engineering 43: Electrical Circuits & Devices
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-43_Prob_5-80_Dual-Src_Complex-Pwr_Solution_.pptx